where \(\nabla f (x^p)\)  is the gradient of \(f\)  at \(x^p\) .
The formula (REF ) is the well-known iterative hard thresholding (IHT) initially studied in [1]}, [2]}.
The IHT can be enhanced by either attaching an orthogonal projection (a pursuit step) to obtain the so-called hard thresholding pursuit (HTP) method [3]}, [4]} or by using an adaptive stepsize strategy to yield the so-called normalized iterative hard thresholding (NIHT) [5]}.
While the algorithm (REF ) can reconstruct the vector under suitable conditions (see [2]}, [7]}, [8]}), but as pointed in [9]}, [10]}, the operator \({\cal H}_k\)  may cause certain numerical problems as well.
